arXiv subjects
Publications and source records attributed to Khyati Sharma.
In [Finite groups with a small number of cyclic subgroups. Czechoslovak Mathematical Journal 75.3 (2025): 839-851], the authors classified all the groups having 13 cyclic subgroups. In this paper, we give a new proof for the classification of all finite groups having exactly 13 cyclic subgroups.
In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially extends the classification of n-cyclic groups for n\geq 13.
For a finite group \(G\), let \(\sub(G)\) be the number of subgroups of \(G\), let \(\cyc(G)\) be the number of cyclic subgroups, and let \(\pi(G)\) be the number of distinct prime divisors of \(|G|\). We study the normalized counts \(\lambda(G)=\sub(G)/2^{\pi(G)}\) and \(\eta(G)=\cyc(G)/2^{\pi(G)}\). We prove that \(\eta(G)<5/4\) or \(\lambda(G)<3/2\) implies that \(G\) is cyclic of squarefree order. The inequalities \(\eta(G)<2\) and \(\lambda(G)<5/2\) each force all Sylow subgroups to be cyclic, and hence imply metacyclicity. For the subgroup count, we give an exact arithmetic criterion in the parameters of the corresponding \(ZM\)-presentation. We determine all values with \(1<\eta(G)<2\) and \(1<\lambda(G)<5/2\), and prove that \(\lambda(G)<59/8\) or \(\eta(G)<4\) implies solvability. Both solvability bounds are sharp. We also describe how cyclic direct factors of coprime order affect the two normalized counts.
A finite group is called a CLT-group if it contains a subgroup corresponding to every divisor of the order of the group. It is said to be a Cyclic (Abelian) CLT group if it contains a cyclic (abelian) subgroup corresponding to every proper divisor of the order of the group. A natural number is said to be a CCLT (ACLT) number if every group of that order is a cyclic (abelian) CLT group. In this work, we classify all CCLT and ACLT numbers and study various properties of Cyclic (Abelian) CLT groups. We also show that the classes of CCLT and ACLT groups are contained in the class of supersolvable groups. Moreover, we introduce the function CCLT-degree on the set of non-cyclic finite groups and study the properties of this function.
A cyclic subgroup graph of a group $G$ is a graph whose vertices are cyclic subgroups of $G$ and two distinct vertices $H_1$ and $H_2$ are adjacent if $H_1\leq H_2$, and there is no subgroup $K$ such that $H_1<K<H_2$. M.T\u{a}rn\u{a}uceanu gave the formula to count the number of edges of these graphs. In this paper, we explore various properties of these graphs.
A finite group is said to be $n$-cyclic if it contains $n$ cyclic subgroups. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group $G$ and is denoted by $cdeg (G)$. In this paper, we classify all $12$-cyclic groups. We also prove that the set of cyclicity degrees for all the finite groups is dense in $[0,1]$, which gives a solution to the problem asked by T\u{a}rn\u{a}uceanu and T\'{o}th in [20] "For every $a\in [0, 1]$, does there exist a sequence $(G_n)$ of finite groups such that $\lim_{n\to\infty} cdeg(G_n)=a$ "?